A) \[{{2}^{20}}{{(-1+i\sqrt{3})}^{20}}\]
B) \[{{2}^{20}}{{(1-i\sqrt{3})}^{20}}\]
C) \[{{2}^{20}}{{(-1-i\sqrt{3})}^{20}}\]
D) None of these
Correct Answer: D
Solution :
Let\[z=-1+i\sqrt{3}\], \[r=\sqrt{1+3}=2\] \[\theta ={{\tan }^{-1}}\left( \frac{\sqrt{3}}{-1} \right)=\frac{2\pi }{3}\] \[\therefore \,z=2\,\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)\] \[\therefore \,\,{{(z)}^{20}}={{\left[ 2\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right) \right]}^{20}}\] \[={{2}^{20}}{{\left( \cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3} \right)}^{20}}\]\[={{2}^{20}}{{\left( -\frac{1}{2}+i\frac{\sqrt{3}}{2} \right)}^{20}}\].
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